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Conformal Mapping

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1. Introduction. Attention will be confined to a group of problems centering around so-called schlicht functions—that is, functions regular in a given domain and assuming no value there more than once. The type of problem we consider involves determination of precise bounds for certain quantities depending on the function/, as ƒ ranges over the schlicht functions in question. Since, for suitable normalization of the functions at some fixed point of the domain, the resulting family of functions is compact or normal, the extremal schlicht functions always exist and the problem is to characterize them. Interest was focused on this category of questions by the work of Koebe in the years 1907-1909, who established for the family of funct i o n s / o f the form ƒ(z) = z+a2Z2+aszz+ • • • , schlicht and regular in \z\ < 1 , a series of properties, among them the theorem of distortion bearing Koebe's name. This theorem asserts the existence of bounds for the absolute value of the derivative ƒ'(s), these bounds depending only on \z\. Further efforts were directed toward finding the precise values of the bounds asserted by Koebe's theorem, but success was not attained until 1916 when Bieberbach, Faber, Pick and others gave a final form to the theorem of distortion. At the same time the precise bound for | a2\ was given, namely 2, and the now famous conjecture was made that \an\ ^n for every n. Since 1916 this group of problems has attracted the attention of many, and there is now a considerable literature. The present state of this sphere of questions will be described briefly in a general sort of way, and a few outstanding problems will be indicated, but no attempt at completeness has been made. 2. The coefficient problem. Let 5 be the family of functions (2.1) ƒ(*) = z + a2z2 + azzz + • • •

which are regular and schlicht in \z\ < 1 . The most famous problem concerning these functions is whether \an\ ^n (n~2, 3, • • • ), with equality for any n only in the case when (2.2) ƒ(*) =
(1 Z


= z + 2Vz* + 3*8» + • • • ,

| n | - 1.

An address delivered before the Los Angeles meeting of the Society on November 30, 1946, by invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings; received by the editors December 30, 1946.





It is this problem which has stimulated much of the research leading to the various methods, in particular the method of parametric representation given by Löwner [7 J1 in 1923 and the recent methods (see [ l l , 12]) in which the extremal function is compared with infinitesimal variations more general than those provided by Löwner's method. There are several short proofs that \an\ ^n (n — 2, 3, • • • ) when all the coefficients are real, and a simple proof, due to Rogosinski [lO], will be sketched here. Without loss of generality we may assume that ƒ is regular and schlicht in \z\ ^ 1 , for every schlicht function is the limit of such functions. Since ƒ has real coefficients, it takes conjugate values for conjugate values of z and so the map of | z\ < 1 by ƒ is symmetrical with respect to the real axis. Since ƒ is schlicht, Im ƒ and Im s have the same sign in \z\ ^ 1 . The function (2.3) l—^-f(z)=p{z) z therefore has a positive real part in \z\ < 1 since on the boundary we have, writing z = ei6, Re p(eid)*=2 sin 0-Im f(ei9)^0. It follows that the coefficients of p are majorized by those of the function 1+z

« 1 + 2s + 2s2 + . . . .

1 — z Since the coefficients of this function as well as the coefficients of the function (2.4) — 1 — 2 = s + *3 + s 5 + . . . 1 — z

are positive, we see at once, multiplying both sides of (2.3) by (2.4), t h a t the coefficients of ƒ are majorized by the coefficients of the function (2.5)
Z iA

(1 - z)2


- z + 2*2 + 3z* + . • . .

Similarly, \an\ ^n when ƒ maps \z\ < 1 on a star-like domain. In both cases equality is attained only when ƒ is of the form (2.2). We see here a connection between schlicht functions and functions with positive real part, a connection that will be further emphasized in §3 below. In the general case \an\ , 2f sin 2 tan a = 1 + 2r cos 2tf> + r2 2 2 Cx -s (1 + r) c o s A ç*2 = (1 ~r) sin $, 2r 2r X = Ci log

/ ( \


w\ < a < — ), 2 2/

+ C 2 a - 2 cos 0,

(1 + r)*
/x = C2 log — - Cia + 2 sin 0. (1 - r) 2 The second analytic surface is defined by 02 = (X2 + /z2)1'2 «3 = X2 + M2 + (Ci + fC«)(X - t/i) \ X — ifx + (r + — + cos 2* Y X + ijit If ƒ(2) belongs to a point on this boundary surface of 5| 0) , then w=/(z) maps \z\ < 1 on the w-plane minus a single curved analytic p x




slit extending from w = to some finite point. As r—»0, the slit tends = * to a straight line arg (w) = constant and the corresponding ƒ tends to z/{l+er»z)K Asr->1, X — 2 cos {log (cos ju — 2(sin — cos 0). »

T h a t is, r = 1 corresponds to the edge of intersection of the two surfaces. For functions w—f(z) belonging to this edge, one of the two prongs of the fork is absent. Even in the case w = 3 w e observe that the equations defining the boundary of 5 3 are sufficiently complicated that it is difficult to infer directly from them t h a t |a 3 | g 3 . Since for n>3 the situation is much more complicated, the precise bounds for the coefficients, a specific question about these regions, remain undetermined. Other questions concerning these regions may be asked; for example, what are the maximum and minimum distances of the boundary of Sn from the origin (distance from the origin being equal to G C ^ I ^ I 2 ) 1 / 2 ). In the case of S 3 , the maximum distance of the boundary from the origin is (2 2 +3 2 ) 1 / 2 = 13 1/2 , attained only for functions of the form (2.2), and the minimum distance of the boundary from the origin is 3 1 / 2 /2, attained only for functions of the form 1 . . yz2 - -v2zz + • • • , U « 1{ l 1 - rjz/21^ + 17V 21'2 2 To each boundary point (a2y aZl • • • , an) of Sn there belongs one and only one function w *=f(z), and this iunction maps the unit circle on the w-plane minus piecewise analytic slits. Given any complex numbers p2, pz, • • • , pn, XX2I P*\2 = 1» let (2.8) = z+ L = Re fad* + pzaz + • • • + pnan) = (p2(t2 + p2&2 + * * ' + pndn + jMn)/2. If the maximum value of L in Sn is Af, then Sn lies entirely on one side of the (2n — 3)-dimensional hyperplane L~M. Since Sn cannot be convex for w > 2 , such a supporting hyperplane can touch the boundary only a t a well-defined subset of boundary points of Sn- Any function w=f belonging to a point of this subset maps \z\ < 1 on the w-plane minus one or more analytic slits meeting at infinity and each of these slits has no finite critical points and is therefore unforked. If for two different sets of numbers £2, pz, • • • , pn the corresponding Us given by (2.9) both have a local extremum at the same boundary point (a2, as, • • • , an) of S n , then the function ƒ belonging to this point is algebraic. These algebraic boundary functions map \z\ < 1 z 1




onto the plane minus slits. It would be interesting to study the character of algebraic schlicht functions which map onto slit regions. The method used to characterize Sn has been sketched in [11(c)] and a detailed exposition is in course of preparation by A. C. Schaeffer and the author. Therefore, no attempt will be made to describe the method here. We remark only that the starting point of the method is to maximize a certain function F(a,2, â2, a3, âz, • • • , any ân) in 5 n , the maximum being attained at a boundary point {a^ a3, • • • , an). Iff belongs to this boundary point, a differential equation f o r / is obtained by making infinitesimal variations of this extremal function. The method then consists of a study of the resulting differential equation. In the next section (§3), Löwner's method of parametric representation will be briefly discussed. This method provides an e-variation of any schlicht function, but the variation is one-sided in that e can have only one sign. This defect does not occur in the more powerful variational methods recently developed (see [ l l ] and [12]). 3. Relations between coefficients of schlicht functions and coefficients of functions with positive real part. In the case of functions with positive real part, the regions of variability of the coefficients have been obtained by Carathéodory [ l ] , Toeplitz [13], and others. If numbers 71, 72, • • • , 7n-i are given, then there is a function (3.1) p(z) =
1 +

2 Ê M '

which is regular and has positive real part in \z\ < 1 with c\ = Yi> £2 = 72, • • • , cn-\ = Yn-i if and only if the Hermitian form n w


H = ]£ ]C YM-^V

is positive semi-definite, where 7o = 1 and 7_fc = 7 * is the complex conjugate of 7*. In geometrical language, there is a function p{z) with Re p(z)>0 in | s | < l if and only if (71, 72, • • • , 7 n -i) lies inside or on the boundary of the smallest convex region containing the curve eie, e2id, • • • , e i(n ~ 1)ô (6 real), and the interior of this region is characterized by the property that H is positive definite there. The nth region of variability Pn is the set of points (ci, c% • • • , cn-i) each of which belongs to some p(z). The points of Pn have the one-to-one parametric representation






~ 0 ^ 0,- < 2ir,

E / * / S 1, y-i EJ-IM/5*!-

and the boundary of Pn is characterized by the condition that T o each boundary point defined by (3.3) there belongs only one function, namely
«—I J JL (>Mjz n—1 E M / /«i



- E M / » *.i 1 -

—> e**z 1.

Rogosinski [lO] has studied the class of functions f(z) = z + «222 + a3z8 + • • • which are regular for \z\ < 1 and have the property that Im ƒ and Im z have the same sign in \z\ < 1 , a property which implies that all the coefficients are real. We have seen that schlicht functions with real coefficients have this property, and the argument used above to prove that \an\ ^n for schlicht functions with real coefficients extends at once to this wider class of power series, a class of functions which Rogosinski has called "typically-real." In fact, if ƒ is typicallyreal then


Ll£.f(z)-p(z) z is a function of positive real part, and conversely. From this simple relation the variability regions Tn for the coefficients of typically-real functions are easily obtained from the corresponding regions Pn for functions with positive real part. I t is readily seen that Tn is the smallest convex region containing the wth variability region for schlicht functions all of whose coefficients are real. This raises an interesting question: what is the smallest convex region containing Sn itself? A domain containing w = 0 is said to be star-like (with respect to w = 0) if any point of it can be joined to the origin by a straight-line segment which lies in the domain. The variability regions S* for the subfamily of schlicht functions which map \z\ < 1 onto star-like domains are also related to the regions Pn in a simple way. This follows from the fact that if ƒ maps | z | < 1 onto a star-like domain, then (3.6) ƒ(*) J-L-piz), zf(z)




where p(z) is regular and has positive real part in \z\ < 1 . Conversely, if p(z) is regular and has positive real part in | s | < 1 , then (3.7) I —~rd^ J o tp{£) is regular in \z\ < 1 and maps the unit circle on a star-like domain. Writing /«-«exp

p{z) - 1 + 2 X ) ^ V , r-l we have



Ù2 =

— 2Ci,

#3 =

~" ^2 +


s a4 = ( - 2c3 + Udc2 - 24*0/3, The parametric representation of Pn given by (3.3) may be used to define a parametric representation of Sn*. Let
(3.9) ah = A*(01, 02, • • • , 0n-lî Ml, M2, * • ' , Mn-l)


be the £th coefficient of the function (3.10) ƒ(*) = — y=i ,

/*, ^ 0, 2 > , ^ 1,

II (1 ~ e**z)** which maps |JS| < 1 on a star-like domain. As the parameters vary over the parameter space, the point (a2, as, • • • , an) sweeps outS w *. The boundary of 5„* is characterized by the condition (3.11) Z \ , = 1, p«i and to each boundary point defined in this way there belongs the unique function given by (3.10). In this case the function (3.10) maps \z\ < 1 on the plane minus q ( l ^ g ^ w — 1) straight-line slits pointing toward the origin, and adjacent slits form at oo an angle equal to 2JJLVTT. The intersection of the boundary of Sn with Sn* is the subset of the boundary of Sf* for which


fxv =



£ w „ = * — 1, v «i

k — 1

where m, and k are integers,






The method of Löwner [7] provides a deeper connection between schlicht functions and functions with positive real part. Heuristically, the method may be briefly described as follows. For each r, O ^ r g T ( T X ) ) , let S(z, r ) , 5(0, r) = 0 , be a schlicht function mapping \z\ < 1 onto a star-like domain D*. Then for any t>0, the values of the function £""*.S(2, r) lie inside D* and so the function $=$(2;, t, r) =5~1(e""'5(2;, r ) , T), where S"1 denotes the function inverse to 5, is regular, schlicht and bounded by 1 in l^j < 1 . We have S(z, r) dt and so (3.13) S-K(r"S{z, t), t)=z^Aràt S (z, t) + o(At).

Now let us define a one-parameter family of functions (3.14) g(z, t) = y(t)(z + a2(t)z* + az(t)z* + • • • ) , y(t) > 0,

which are regular and schlicht in \z\ < 1 for O ^ ^ T and such that (3.15) By (3.13) dg(z, t) g(z, t + At) = g(z91) - 2 - ^ — - p(z, t)At + o(At) dz where p(z, t)=S(z, t)/(zS'(z, t)) has positive real part in \z\ < 1 . Dividing by At and letting At approach zero, we obtain the differential equation (3.16) (3.17) — - — = - z — - — p ( z , t). dt dz g(z, t + At) = g(S-Ke-AtS(z, /), /), t).

If we divide both sides of (3.17) by z and then take 2 = 0, we have
( 3 .18)


„ _





The equation (3.17) may also be given a direct geometrical interpretation. Suppose t h a t g(z, t) and p{z, t) are both regular and Re p>0 in | z | ^ 1 for O g / ^ T . Taking 3 = e" in (3.16) we have




dg(eie, t) (3.19) g(é*, t + At) = g(e*t t) + i p(e", t)At + o(At). dB The map D% of | ^ | < 1 by g(z, /) is a domain bounded by an analytic curve; the quantity idg(ei$, t)/dO has the direction of the inner normal to the boundary of Dt at the point w = g(eie, t) and idg{eiB, t)/dd 'p(eie, t) is a vector which makes an angle less than TT/2 with this inner normal. Equation (3.19) states that the boundary point g(ei9, t+At) of the domain Dt+At lies inside Du Thus, as t increases from 0 to T, the domain Dt shrinks and, if t' (the map of \z\ < 1 by g(z, /'))• Actually, Löwner's parameter t moves in the opposite sense (corresponding to (3.24)), in which case the inclusion relation between D%* and Dt" is reversed, and he showed that any schlicht function of an everywhere dense set can be connected to the function f(z)=z by a curve corresponding to a function p(z, t) of the form (3.25) which depends continuously on /. If the functions p(z, t) are not restricted to be of the form (3.25) and if the continuity in / is dropped, then 3 it can be shown that any schlicht function can be connected to f(z}—z9 but it is not known t h a t this can be done using only functions of the form (3.25), even without continuity. Take / " = r , *' = 0 in (3.24), and let the value T of the parameter correspond tof(z) =z, in which case ak(T)=0 (& = 2, 3, • • • ). Writing ajb(O) =dk (k = 2, 3, • • • ), we then obtain, by recursion, the formulas : (3.26) (3.27) a2 = - 2 f e-TCi(T)dT,

a3 = - 2 f

e-^c2{T)dr + 4 ( f




e-ZTcz{r)dr + 12 I e~ 2r c 2 (r)Jr- | e-TCi{r)dr •/ o Jo

• T /» r

- 8 ( ƒ e-Mr)dr J ,

An unpublished result of A. C. Schaeffer and the author.


/ J \ IJ J- a i « a • • • ajfe



••• I


— X) avTv

* I I Cav{rv)dTldT2 • • • ^T*, v=l where r a i ^ . . . « f c = 2 A (^ — «i)(n — ai — a 2 ) • • • (n — ai — « 2 — • • • — «*), the ai, a 2 , • • • , a& being positive integers with sum n — l. Now the interior points of Sn are characterized by the property that bounded functions belong to them. Given an interior point (a2, a3, • • • , a») of Sn, let the minimum maximum modulus of all functions ƒ belonging to this point be e'(/^Q). The set of points of Sn which are representable by the formulas (3.26) to (3.29), in which the CV(T) 0> = 1, 2, • • • , n — 1) are measurable functions of T, is exactly the set of interior points of Sn corresponding to the values t^T. If, therefore, we take T= oo in these formulas, any point of 5 w may be given this integral representation for suitable choice of the curve (ci(r), £ 2 (r), • • • , cn-i(r)). More generally, given any schlicht function f(z) = z + a2z2 + azzz + • • • , there is a function p(z, r) = 1+2^^>:SS1CV(T)ZV such that the coefficients a n (« = 2, 3, • • • ) are given by (3.29). Conversely, given any function p(z, r ) , the coefficients denned by (3.29) belong to a schlicht function. The correspondence between f unctions ƒ (2) and p(z, r) is not one-to-one; in general infinitely many p(z, r) correspond to a schlicht function ƒ. To any ƒ which has real coefficients, there is a corresponding p(z, r) which has real coefficients. Now if Cu £2, £3, • • • belong to a function with positive real part, so do the numbers Re £1, Re c2, Re cz, • • • (this follows from the convexity of the family of functions p). Thus if in (3.29) we replace the c„(r) by their real parts, the resulting an belongs to a schlicht function with real coefficients. From this remark, it follows at once that |a 2 | r§2, \az\ g 3 , for complex coefficients. For example, to show that |a 3 | ^ 3 , we may suppose without loss of generality t h a t a 3 > 0 . For, given any schlicht function ƒ, we have only to consider the function e~i9f(ei9z) = z + a2ei$z2 + aze™zz + • • • which, for suitable choice of 0, will have a real non-negative third co-




efficient. But if az is real we have from (3.27) (with T = oo) a3 = - 2 J - 4M e~2r Re

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